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University & Postgraduate study notes
Use these for econometrics, statistics, time series, finance and dissertation-methods support.
- Introduction to Econometrics: a free course, lesson by lesson (18 lessons published)
- Econometrics by stage: the course, marked questions, explainers and lecture notes
- Econometrics basics, OLS, Law of Large Numbers & Central Limit Theorem
- Mathematical methods (linear algebra & calculus) study notes
- Time series econometrics materials
- Cross-section econometrics materials
- Econometric methods materials
- Postgraduate time-series materials
- Financial econometrics materials
A-Level, admissions and TMUA resources are on the Resources page.
- A-Level & introductory statistics study notes, probability, distributions & hypothesis testing
Archive note: some downloadable materials are historic notes written for previous teaching and do not represent any current course or current course content.
EC3304 · University of St Andrews · Study-notes hub · linear algebra & multivariate calculus
Complete study notes for the mathematical toolkit of economics: vectors and matrices, linear systems, eigenvalues and quadratic forms, then the implicit function theorem, Hessians and concavity, Taylor series and integration, written up with worked examples, fully solved practice problems and downloadable lecture slides. Open the study notes hub →
Study notes & downloads
Open notes
Mathematical methods
Mathematical Methods for Economics
Topics covered
Study-notes hub covering the mathematics behind economic theory: linear algebra (vectors, matrices, linear systems, determinants, rank, eigenvalues) and multivariate calculus (implicit function theorem, Hessian and concavity, mean value theorem and Taylor series, integration and Leibniz's rule). Includes worked examples throughout and a set of fully solved practice problems spanning both halves, from determinants to Kuhn-Tucker optimisation.
ECON30401 · University of Manchester · Third-year undergraduate · Historic teaching archive, lectures 1-4
Univariate time series modelling from first principles. These archived materials cover the first part of the time-series material. Later topics such as VAR systems, unit roots and cointegration are not included here.
New: a full set of time series econometrics study notes, stationarity and ARMA, estimation and inference, hypothesis testing, forecasting and seasonality, and an introduction to VAR system modelling, each written up with derivations, worked examples and self-check questions, plus exercises and an EViews computer lab.
Lecture slides
Lecture 1 · 2019 version (latest)
Introduction to Time Series Econometrics
Topics covered
Time series process vs. realisation. Strict and covariance stationarity formally defined. White noise. MA(q) process, definition, mean, variance, autocovariance function. Sample realisations of AR, MA and ARMA processes. Sets up all notation used in subsequent lectures.
Lecture 2 · 2016 version (latest)
ARMA(p,q) Processes & Stationarity
Topics covered
MA(∞) process and squared/absolute summability. Wold Decomposition Theorem. AR(1) as MA(∞), mean α/(1−φ₁), variance σ²/(1−φ₁²), ACF ρ(k)=φ₁ᵏ. ARMA(1,1) properties. Lag operator introduced. General ARMA(p,q). Stationarity via characteristic polynomial, roots of 1−φ₁L−…−φₚLᵖ must lie outside the unit circle.
Lecture 3 · 2015 version
Estimation, Hypothesis Testing & Model Selection
Topics covered
Sample ACF: ρ̂(k) = Σ(Yₜ−Ȳ)(Yₜ₋ₖ−Ȳ)/Σ(Yₜ−Ȳ)². Under H₀: ρ(k)=0, √T·ρ̂(k) →ᵈ N(0,1). Correlogram and 95% bands ±1.96/√T. Ljung-Box portmanteau test Q(m)~χ²(m). AR estimation by OLS; MA and ARMA by MLE. AIC = −2·log(L̂)+2K; BIC = −2·log(L̂)+K·log(T). Applied to UK GDP growth data.
Lecture 4 · 2015 version
Prediction & Seasonality
Topics covered
AR(1) one-step predictor Ŷₜ₊₁=µ+φ₁yₜ. J-step ahead prediction and forecast error variance σ²·Σφ₁²⁽ᴶ⁻ʲ⁾. Long-horizon convergence to unconditional mean. Deterministic seasonality: seasonal dummies, dummy variable trap, seasonal adjustment by OLS residuals. Stochastic seasonality: SAR(1), its MA(∞) form, ACF. Combined model with dummies and AR dynamics. UK retail sales application.
Written notes
Available on request
Weeks 1-2 · Comprehensive notes (final version)
Time Series Econometrics. Lecture Notes Weeks 1 & 2
Topics covered
Written prose notes (not slides) covering the full material from Lectures 1-3. Full derivations of MA(q) and AR(1) properties, Wold decomposition proof, ACF and PACF theory, sample ACF distribution under the null, model estimation rationale, information criteria derivation. Suitable for reading alongside or instead of slides. Best reference for exam revision.
Foundation · OLS and linear models
OLS Notes
Topics covered
The linear model used throughout econometrics: OLS estimator, Gauss-Markov theorem, large-sample properties and inference.
Exercises & solutions
PDF
Exercise 1 · ARMA properties
Problem Set 1: ARMA Properties & Wold Decomposition
Topics covered
Moments and autocorrelations of an MA(2), stationarity and moments of an AR(2), strict against weak stationarity (a Cauchy process), and the consistency of the sample mean. Four tutorial questions.
Video Exercise 2 · AR(2) and ARMA(1,1)
Video Exercise 2: AR(2) Prediction & ARMA(1,1) Moments
Topics covered
Deriving 1, 2 and 3-step ahead forecasts and forecast error variances for AR(2). Deriving mean, variance and ACF for ARMA(1,1) via the MA(∞) route. Supported by video recordings.
Class Exercise 2 · IBM stock returns
Applied Exercise: ARMA Modelling of IBM Monthly Returns
Topics covered
IBM monthly stock returns 1991-2013 (T=276). Correlogram examination, white noise testing, model identification, estimation, diagnostics. Requires interpreting EViews output.
PC Lab 1 · EViews, UK real GDP
PC Lab 1: Time Series in EViews. UK Real GDP 1955-2013
Topics covered
Hands-on EViews exercise. UK quarterly real GDP (seasonally adjusted). Plot, growth rates, ACF/PACF, ARMA estimation, information criteria for model selection, forecasting, residual diagnostics.
Later topics such as VAR systems, unit roots and cointegration are not included here.
ECON61001 · University of Manchester · Postgraduate · Historic teaching archive, lectures 5-7
Diagnostic testing, endogeneity and instrumental variables, the three topics most relevant to applied econometric research. These lectures cover material central to any serious empirical economics paper.
Lecture slides
Lecture 5 · Diagnostic testing
Diagnostic Testing & Robust Inference
Topics covered
Testing OLS assumptions in practice. Heteroskedasticity: White test (T·R² ~χ²), Breusch-Pagan; heteroskedasticity-consistent (HC) standard errors. Serial correlation: Durbin-Watson, Breusch-Godfrey LM test; HAC (Newey-West) standard errors. 2016 version.
Lecture 6 · Instrumental variables
Endogeneity & Instrumental Variables (2SLS)
Topics covered
When E[uᵢxᵢ]≠0. OLS biased and inconsistent. Sources: simultaneity, omitted variables, measurement error. IV estimator. 2SLS: first stage regresses xᵢ on zᵢ; second stage uses x̂ᵢ. Validity conditions: relevance (Cov(zᵢ,xᵢ)≠0) and exclusion restriction (Cov(zᵢ,uᵢ)=0). First-stage F>10 rule of thumb. Acemoglu-Johnson-Robinson settler mortality example. 2016 version.
Lecture 7 · Weak instruments
Instrumental Variables with Weak / Unidentified Instruments
Topics covered
Weak instrument problem: 2SLS severely biased toward OLS; standard Wald confidence intervals have severe size distortions. Concentration parameter. Stock-Yogo (2005) critical values (F>16.38 for 5% maximal size distortion with one instrument). Anderson-Rubin identification-robust test. LIML as less biased alternative to 2SLS. Conditional likelihood ratio (CLR) test. Directly connected to Dr Grant's CeMMAP research on singular moment variance.
Tutorial problem sets & solutions
Worked exercises
Tutorials 3-4 · WLS and GLS (worked on the page)
Tutorials 3 & 4: Weighted Least Squares & Generalised Least Squares
Topics covered
Deriving the WLS estimator, properties under heteroskedasticity, GLS as efficient estimator when errors are non-spherical. Three worked exercises with full solutions, on the page.
Tutorial 5 · Robust inference
Tutorial 5: Heteroskedasticity & Serial Correlation. Applied Problems
Topics covered
White test for heteroskedasticity, linear-form test, HC standard errors in practice. Serial correlation testing, Newey-West HAC standard errors. Interpretation of regression output with diagnostics. Questions 1 and 2 with worked solutions (question 3 and the video questions, which were a colleague's, are not included).
The OLS notes linked above cover the foundations needed before the diagnostic testing and instrumental variables material.
ECON60052 · University of Manchester · Postgraduate · Cross-section & panel econometrics, worked study notes
Causal inference and applied microeconometrics: treatment effects and the potential-outcomes framework, robust and cluster-robust inference, difference-in-differences policy analysis, and the fixed- and random-effects panel estimators. Each lecture is a full worked study page with self-check questions. Open the hub →
Study notes & lecture slides
Study notes
Lecture 1 · Causality
Potential outcomes, ATE and ATT, the selection-bias decomposition, randomisation, unconfoundedness, overlap and matching estimators.
Lecture 2 · Inference
White/Eicker-Huber robust standard errors, testing for heteroskedasticity, the intraclass correlation, the Moulton factor and cluster-robust inference.
Lecture 3 · Policy
Pooled cross sections, interaction terms, difference-in-differences, natural experiments, common trends and two-period first differencing.
Lecture 4 · Panel
First-difference, fixed-effects (within/LSDV) and random-effects estimators, the FE-FD comparison, the Hausman test and Mundlak's correlated random effects.
EC5221 · University of St Andrews · Postgraduate · Historic teaching archive
Postgraduate-level time series covering the same core theory as the undergraduate course but with greater mathematical rigour, formal asymptotic arguments, and extension to VAR systems. Latest versions are from 2020/2021.
Lecture slides
Week 1 · 2021 version (latest)
Stationary Univariate Time Series. ARMA Processes & Stationarity
Topics covered
Covers the full formal treatment: time series process vs. realisation, strict and covariance stationarity, bivariate moments and the autocovariance function, white noise, ARMA(p,q), moments and stationarity conditions via the lag operator, MA(∞) and Wold decomposition, AR(1) as MA(∞). More formal and complete than the undergraduate version of the same material.
Week 2 · 2021 version (latest)
Stationary Univariate Time Series. Estimation & Inference
Topics covered
Sample moments: sample mean, sample ACF and their asymptotic distributions. Statistical properties of sample ACF under H₀ and H₁. Portmanteau tests. Maximum likelihood estimation of ARMA processes: Gaussian log-likelihood, conditional vs. exact MLE, Newton-Raphson. AIC and BIC for order selection. Reading list and applications.
Week 5 · Multivariate time series
Topics covered
The VAR(p) model and its vector moving-average (Wold) representation. Impulse response functions as the MA coefficient matrices. The problem of contemporaneously correlated reduced-form shocks, orthogonalisation via the Cholesky decomposition, orthogonalised impulse responses and the identification/ordering issue. Worked study notes plus slides.
Problem sets & solutions
Problem Set 2 · VAR systems · 2021 version (latest)
Problem Set 2: Bivariate VAR(1) Systems
Topics covered
Bivariate VAR(1): Yₜ = α + Φ₁Yₜ₋₁ + εₜ. Stationarity condition via eigenvalues of Φ₁. Mean, variance and autocovariance of stationary VAR(1). Impulse response functions. Granger causality testing. With specific parameter values (φ₁₁=0.6, φ₁₂=0.1, φ₂₁=−0.6, φ₂₂=1.1).
EC5609 · University of St Andrews · Postgraduate · Historic teaching archive
The statistical properties of financial data, ARMA modelling of returns, maximum likelihood, and advanced volatility modelling with GARCH. Applied throughout with stock return data.
Lecture slides
Lecture 1 · 2021/22 version (latest)
Distribution Theory & Properties of Stock Price Returns
Topics covered
Financial econometric data. Log returns rₜ=log(Pₜ)−log(Pₜ₋₁). Statistical distribution theory: PDF, CDF, moments. Sample mean, variance, skewness (third standardised central moment), kurtosis (fourth). Leptokurtosis in returns: excess kurtosis>0, fat tails. Normal vs. Student-t. Central Limit Theorem and asymptotic normality of sample moments. Sampling distributions. Motivation for ARMA and GARCH modelling of returns.
Lecture 7 · 2021/22 version (latest)
Advanced Volatility Modelling. ARCH & GARCH
Topics covered
Conditional heteroskedasticity in financial returns. ARCH(q): hₜ=ω+Σαᵢεₜ₋ᵢ². GARCH(1,1): hₜ=ω+α·εₜ₋₁²+β·hₜ₋₁; stationarity requires α+β<1. Volatility clustering. Asymmetric effects (leverage): EGARCH (Nelson) log(hₜ)=ω+α·|εₜ₋₁/√hₜ₋₁|+γ·εₜ₋₁/√hₜ₋₁+β·log(hₜ₋₁). TARCH/GJR-GARCH. GARCH-in-mean. Generalised error distribution for fat tails. MLE of GARCH models. Empirical examples.
Maximum likelihood · supplementary notes
Maximum Likelihood Estimation in Financial Econometrics
Topics covered
Why maximum likelihood: non-linear models (MA and GARCH) that OLS cannot fit; the idea of ML through the Kullback-Leibler distance between the assumed and the true density; the maximum likelihood estimator defined; weekly Apple returns and their correlograms as the running example. 23 slides.
Problem sets & solutions
PDF
Problem Set 1 · 2023/24 version (latest)
Problem Set 1: Applied. US Stock Market Returns (MktRF)
Topics covered
Applied problems using Stata output for MktRF, weekly excess US stock market returns (%) March 2000 to May 2021 (T=255). Statistical properties, ARMA modelling, diagnostics. Most recent version of the course problem set.
Problem Set 1 · 2021/22 version + solutions
Problem Set 1: Distribution Theory. Uniform, Normal, Student-t
Topics covered
Theoretical problems. Uniform X~Unif[0,1]: mean, variance, skewness, kurtosis. AR(1) and MA(1) properties review. Moment estimation from financial data. Full worked solutions available.
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Turn the notes into marks
These PDFs come from my own university lectures, and the fastest way through a module that isn't clicking is to work the material one-to-one. Send the module title, topic list and deadline; a free 30 minute consultation identifies where support pays off most.
Study notes from the time-series material
Time series notes
broken into linked explanations
The study notes below separate the concepts into clearer linked explanations: process and realisation, stationarity, white noise, autocorrelation, correlograms, ARMA models, lag-operator derivations and estimation.
Core conceptsProcess, realisation, stationarity, white noise and ACFBest starting point for students new to time series.CorrelogramSample ACF, Q-tests and residual serial correlationThe practical diagnostic page linked from the other notes.ARMA modelsMA(q), AR(1), AR(p), ARMA(p,q)Intermediate model interpretation.Advanced derivationsLag operators, roots, Wold decomposition and MLClosest to the advanced TeX derivations.
Studying one of these topics?
For university or postgraduate support, send the module title, topic list, deadline and the specific problem area. For school or admissions support, use the A-Level & Admissions.
Teaching videos · YouTube @unientryTMUA
Teaching videos and worked econometrics lectures
The @unientryTMUA YouTube channel contains free worked videos in econometrics and time series. Use them to preview the style of explanation before sending a university or postgraduate enquiry.
PlaylistTime Series EconometricsStationarity, unit roots, ARIMA, autocorrelation, Newey-West and cointegration.
Channel playlistsApplied Econometrics VideosRegression, inference, IV, heteroskedasticity, autocorrelation and worked examples.
Time Series Econometrics · 5 videos
Covers: stationarity, unit roots, ADF tests, spurious regression, ARIMA models, autocorrelation, Newey-West standard errors, cointegration. Read companion article →
Applied econometrics video topics
Short worked videos covering regression theory, matrix methods, asymptotics, time series and applied techniques. The list below is collapsed so you can scan the main themes without losing the page flow.
View topics covered
∑Gauss-Markov Theorem, sketch of a proof
∑Two-tailed regression t-test
∑Regression coefficient confidence intervals
∑Multicollinearity
∑p-values
∑F-test for linear restrictions
∑IV notation and motivation
∑Matrix algebra: addition, multiplication, inverse
∑The regression model in matrix form
∑Estimator consistency and laws of large numbers
∑Asymptotic normality of OLS estimators
∑Random regressors and OLS properties
∂Regression assumptions for time series data
∂Spurious regression
∂ARIMA estimation in R
∑Introduction to heteroskedasticity
∑Generalised least squares (GLS) theory
∑Detecting heteroskedasticity
∑Heteroskedasticity detection in EViews
∂Introduction to autocorrelation
∂Newey-West standard errors
∑Ramsey RESET test
∑Using dummy variables to model structural change
∑Measurement error
∑Two stage least squares to IV estimator
∂Testing for autocorrelation
∑Structural change example
∑IV and proxy variables
∑Maximum likelihood, a Poisson example
∑Introduction to Bayesian econometrics
∑Exam questions and worked solutions (multiple)
Visit the @unientryTMUA YouTube channel or open the playlist page.
Go deeper
Related resources
on this site
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Time series guide, unit roots, cointegration, ARIMA
Plain-English guide to the topics in Lectures 5-8 of the undergraduate course (not included above). Unit root tests, ARIMA, GARCH, cointegration.
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What is OLS? The foundation for all material above
All four courses assume knowledge of OLS. This guide explains the estimator, Gauss-Markov theorem, and inference from first principles.
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Econometrics explained, a plain-English topic guide
A structured tour of the core econometrics topics, from OLS and inference to time series and instrumental variables, linking back to the worked notes above.
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Teaching videos, 30+ applied econometrics lectures
The @unientryTMUA channel covers applied econometrics across all levels. Free to watch. Complements the lecture notes above.
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Ask a question, forums and free help
r/econometrics, Cross Validated, TheStudentRoom, Economics Stack Exchange, where to get free help, and where Dr Grant answers questions.
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Free 30 minute initial consultation. All topics in the notes above, and beyond, undergraduate through PhD.